22: The Root test leads to
Since this constant is smaller than 1, the investigated series and therefore also the given series converge.
The limit from the Ermakoff test can be done relatively easily using the calculations done before for the Root test as applied to the given series.
There are other alternatives. For instance, we know that logarithms grow
slower than powers, in particular
Or one can change everything into an exponential
and then show that the expression in the exponent goes to negative infinity, there are quite a few ways to get this result.
Anyway, r is less than 1, which confirms convergence of the given series.